WASHINGTON — Linking global warming to public health, disease and extreme weather, the Obama administration pressed ahead Friday with tough requirements to limit carbon pollution from new power plants, despite protests from industry and Republicans that it would dim coal’s future. The proposal, which would set the first national limits on heat-trapping pollution from future power plants, is intended to help reshape where Americans get electricity, moving from a coal-dependent past into a future fired by cleaner sources of energy. It’s also a key step in President Barack Obama’s global warming plans, because it would put in motion plans to end what he called “the limitless dumping of carbon pollution” from all power plants. Under the law once the Environmental Protection Agency controls carbon at new plants, it will control carbon at existing plants, a regulation the agency said Friday it would start work on immediately to meet a June 2014 deadline. Yet the federal government’s own analysis of the new power plant proposal concludes that it would have a “negligible” impact on carbon dioxide emissions, pose little to no costs for the industry and provide no additional benefits to the public by 2022. That’s because it essentially locks in what was widely expected to happen anyway. Even without new federal regulations, the agency concluded that no new coal plants would have been built. Instead, the bulk of new power in this country would be supplied by natural gas, which already meets the standard announced Friday.